Improved Inapproximability for TSP Michael Lampis KTH Royal Institute of Technology August 15, 2012
The Traveling Salesman Problem Input: • An edge-weighted graph G ( V, E ) Objective: • Find an ordering of the vertices v 1 , v 2 , . . . , v n such that d ( v 1 , v 2 ) + d ( v 2 , v 3 ) + . . . + d ( v n , v 1 ) is minimized. • d ( v i , v j ) is the shortest-path distance of v i , v j on G Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
The Traveling Salesman Problem Improved Inapproximability for TSP – APPROX 2012 2 / 16
TSP Approximations – Upper bounds • 3 2 approximation (Christofides 1976) For graphic (un-weighted) case • 3 2 − ǫ approximation (Oveis Gharan et al. FOCS ’11) • 1 . 461 approximation (M¨ omke and Svensson FOCS ’11) • 13 9 approximation (Mucha STACS ’12) • 1 . 4 approximation (Seb¨ o and Vygen arXiv ’12) Improved Inapproximability for TSP – APPROX 2012 3 / 16
TSP Approximations – Lower bounds • Problem is APX-hard (Papadimitriou and Yannakakis ’93) • 5381 5380 -inapproximable (Engebretsen STACS ’99) • 3813 3812 -inapproximable (B¨ ockenhauer et al. STACS ’00) • 220 219 -inapproximable (Papadimitriou and Vempala STOC ’00, Combinatorica ’06) Improved Inapproximability for TSP – APPROX 2012 4 / 16
TSP Approximations – Lower bounds • Problem is APX-hard (Papadimitriou and Yannakakis ’93) • 5381 5380 -inapproximable (Engebretsen STACS ’99) • 3813 3812 -inapproximable (B¨ ockenhauer et al. STACS ’00) • 220 219 -inapproximable (Papadimitriou and Vempala STOC ’00, Combinatorica ’06) This talk: Theorem 185 There is no 184 -approximation algorithm for TSP , unless P=NP . Improved Inapproximability for TSP – APPROX 2012 4 / 16
Reduction Technique We reduce some inapproximable CSP (e.g. MAX-3SAT) to TSP . Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique First, design some gadgets to represent the clauses Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique Then, add some choice vertices to represent truth assignments to variables Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique For each variable, create a path through clauses where it appears positive Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique . . . and another path for its negative appearances Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique A truth assignment dictates a general path Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique We must make sure that gadgets are cheaper to traverse if corresponding clause is satisfied Improved Inapproximability for TSP – APPROX 2012 5 / 16
Reduction Technique For the converse direction we must make sure that ”cheating” tours are not optimal! Improved Inapproximability for TSP – APPROX 2012 5 / 16
How to ensure consistency • Papadimitriou and Vempala design a gadget for Parity. • They eliminate variable vertices altogether. • Consistency is achieved by hooking up gad- gets ”randomly” • In fact gadgets that share a variable are connected according to the structure dic- tated by a special graph • The graph is called a ”pusher”. Its ex- istence is proved using the probabilistic method. Improved Inapproximability for TSP – APPROX 2012 6 / 16
How to ensure consistency • Basic idea here: consistency would be easy if each variable occurred at most c times, c a constant. • Cheating would only help a tour ”fix” a bounded number of clauses. Improved Inapproximability for TSP – APPROX 2012 7 / 16
How to ensure consistency • Basic idea here: consistency would be easy if each variable occurred at most c times, c a constant. • Cheating would only help a tour ”fix” a bounded number of clauses. • We will rely on techniques and tools used to prove inapproximability for bounded-occurrence CSPs. • Main tool: an ”amplifier graph” construction due to Berman and Karpinski. Improved Inapproximability for TSP – APPROX 2012 7 / 16
How to ensure consistency • Basic idea here: consistency would be easy if each variable occurred at most c times, c a constant. • Cheating would only help a tour ”fix” a bounded number of clauses. • We will rely on techniques and tools used to prove inapproximability for bounded-occurrence CSPs. • Main tool: an ”amplifier graph” construction due to Berman and Karpinski. • Result: an easier hardness proof that can be broken down into independent pieces, and also gives an improved bound. Improved Inapproximability for TSP – APPROX 2012 7 / 16
Overview We start from an instance of MAX-E3-LIN2. Given a set of linear equations (mod 2) each of size three satisfy as many as possible. Known to be 2-inapproximable (H˚ astad). Improved Inapproximability for TSP – APPROX 2012 8 / 16
Overview We use the Berman-Karpinski amplifier construction to obtain an instance where each variable appears exactly 5 times (and most equations have size 2). Improved Inapproximability for TSP – APPROX 2012 8 / 16
Overview Improved Inapproximability for TSP – APPROX 2012 8 / 16
Overview A simple trick reduces this to the 1in3 predicate. Improved Inapproximability for TSP – APPROX 2012 8 / 16
Overview From this instance we construct a graph. Improved Inapproximability for TSP – APPROX 2012 8 / 16
Overview From this instance we construct a graph. Rest of this talk: some more details about the construction. Improved Inapproximability for TSP – APPROX 2012 8 / 16
1in3-SAT Input : A set of clauses ( l 1 ∨ l 2 ∨ l 3 ) , l 1 , l 2 , l 3 literals. Objective : A clause is satisfied if exactly one of its literals is true. Satisfy as many clauses as possible. • Easy to reduce MAX-LIN2 to this problem. • Especially for size two equations ( x + y = 1) ↔ ( x ∨ y ) . • Naturally gives gadget for TSP • In TSP we’d like to visit each vertex at least once, but not more than once (to save cost) Improved Inapproximability for TSP – APPROX 2012 9 / 16
TSP and Euler tours Improved Inapproximability for TSP – APPROX 2012 10 / 16
TSP and Euler tours Improved Inapproximability for TSP – APPROX 2012 10 / 16
TSP and Euler tours Improved Inapproximability for TSP – APPROX 2012 10 / 16
TSP and Euler tours • A TSP tour gives an Eulerian multi-graph com- posed with edges of G . • An Eulerian multi-graph composed with edges of G gives a TSP tour. • TSP ≡ Select a multiplicity for each edge so that the resulting multi-graph is Eulerian and total cost is minimized • Note : no edge is used more than twice Improved Inapproximability for TSP – APPROX 2012 10 / 16
Gadget – Forced Edges We would like to be able to dictate in our construction that a certain edge has to be used at least once. Improved Inapproximability for TSP – APPROX 2012 11 / 16
Gadget – Forced Edges If we had directed edges, this could be achieved by adding a dummy intermediate vertex Improved Inapproximability for TSP – APPROX 2012 11 / 16
Gadget – Forced Edges Here, we add many intermediate vertices and evenly distribute the weight w among them. Think of B as very large. Improved Inapproximability for TSP – APPROX 2012 11 / 16
Gadget – Forced Edges At most one of the new edges may be unused, and in that case all others are used twice. Improved Inapproximability for TSP – APPROX 2012 11 / 16
Gadget – Forced Edges In that case, adding two copies of that edge to the solution doesn’t hurt much (for B sufficiently large). Improved Inapproximability for TSP – APPROX 2012 11 / 16
1in3 Gadget Let’s design a gadget for ( x ∨ y ∨ z ) Improved Inapproximability for TSP – APPROX 2012 12 / 16
1in3 Gadget First, three entry/exit points Improved Inapproximability for TSP – APPROX 2012 12 / 16
1in3 Gadget Connect them . . . Improved Inapproximability for TSP – APPROX 2012 12 / 16
1in3 Gadget . . . with forced edges Improved Inapproximability for TSP – APPROX 2012 12 / 16
1in3 Gadget The gadget is a con- nected component. A good tour visits it once. Improved Inapproximability for TSP – APPROX 2012 12 / 16
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